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## Students' Spatial Structuring of 2D Arrays of Squares

We define spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. It is an essential mental process underlying students' quantitative dealings with spatial situations. In this article, we examine in detail students' structuring and enumeration of 2-dimensional (2D) rectangular arrays of squares. Our research indicates that many students do not “see”the row-by-column structure we assume in such arrays. We describe the various levels of sophistication in students' structuring of these arrays and elaborate the nature of the mental process of structuring.

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## Effects of Three Interventions on Children's Spatial Structuring and Coordination of Area Units

We examine the effects of 3 interventions designed to support Grades 2–5 children's growth in measuring rectangular regions in different ways. We employed the microgenetic method to observe and describe conceptual transitions and investigate how they may have been prompted by the interventions. We compared the interventions with respect to children's learning and then examined patterns in observable behaviors before and after transitions to more sophisticated levels of thinking according to a learning trajectory for area measurement. Our findings indicate that creating a complete record of the structure of the 2-dimensional array—by drawing organized rows and columns of equal-sized unit squares—best supported children in conceptualizing how units were built, organized, and coordinated, leading to improved performance.

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## Developing Spatial Structuring with Quick Images

Second graders began their journey to multiplicative reasoning by using rectangular arrays to find a total amount.

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## The Importance of Spatial Structuring in Geometric Reasoning

Second-grader Katy was shown that a plastic inch square was the same size as one of the indicated squares on the seven-by-three-inch rectangle displayed in figure 1a. She was then asked to predict how many plastic squares she would need to cover the rectangle completely.

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## Call for Manuscripts: Spatial Abilities in the Middle Grades

What meaningful spatial-reasoning tasks do you incorporate into your mathematics program? How can teachers help students develop spatial abilities? Why is it important to do so? What are the subtle distinctions, if any, among such terms as the following: spatial sense, spatial visualization, spatial perception, spatial structuring, spatial memory, spatial reasoning, and visual imagery?

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## Fifth Graders' Enumeration of Cubes in 3D Arrays: Conceptual Progress in an Inquiry-Based Classroom

In this study I utilize psychological and sociocultural components of a constructivist paradigm to provide a detailed analysis of how the cognitive constructions students make as they enumerate 3D arrays of cubes develop and change in an inquiry-based problem-centered mathematics classroom. I describe the classroom work of 3 pairs of 5th graders on an instructional activity involving predicting the number of cubes that fit in graphically depicted boxes, and I carefully explicate how the mental processes of abstraction, reflection, perturbation, spatial structuring, and coordination, along with face-to-face social interaction within pairs, brought about meaningful and powerful student learning.

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## Assessing the Development of Preschoolers' Mathematical Patterning

The development of patterning strategies during the year prior to formal schooling was studied in 53 children from 2 similar preschools. One preschool implemented a 6-month intervention focusing on repeating and spatial patterns. An interview-based Early Mathematical Patterning Assessment (EMPA) was developed and administered pre- and postintervention, and again following the 1st year of formal schooling. The intervention group outperformed the comparison group across a wide range of patterning tasks at the post- and follow-up assessments. Children from the intervention group demonstrated greater understanding of unit of repeat and spatial structuring, and most were also able to extend and explain growing patterns 1 year later. In contrast, most of the comparison group treated repeating patterns as alternating items and rarely recognized simple geometrical patterns. The findings indicate a fundamental link between patterning and multiplicative reasoning through the development of composite units.

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## Selecting GeoGebra Applets for Learning Goals

applet gives students the opportunities to construct and coordinate multiple levels of composite units, such as arrays and layers of unit cubes. Such experiences are essential to helping students build a spatial structure that can be used to find the